Introduction To Normed Star-algebras And Their Representations

Introduction to Normed ∗-Algebras and their Representations, 5th ed. arXiv:math.RT/0701306v2 10 Jan 2007 Marco Thill Contents Preface Part 1. v Spectral Theory of Banach ∗-Algebras 1 Chapter 1. Basic Properties of the Spectrum § 1. ∗-Algebras and their Unitisation § 2. Normed ∗-Algebras and their Unitisation § 3. The Completion of a Normed Algebra § 4. The Spectrum § 5. The Spectral Radius Formula: rλ ( · ) § 6. Properties of rλ ( · ) § 7. The Pt´ak Function: rσ ( · ) § 8. Automatic Continuity § 9. Square Roots of Invertible Elements § 10. The Boundary of the Spectrum § 11. Hermitian Banach ∗-Algebras § 12. An Operational Calculus § 13. Odds and Ends: Questions of Imbedding 3 3 7 11 13 17 20 23 26 29 32 34 38 40 Chapter § 14. § 15. § 16. 2. The Gelfand Transformation Multiplicative Linear Functionals The Gelfand Topology ˇ The Stone-Cech Compactification 43 43 49 53 Chapter § 17. § 18. § 19. § 20. § 21. 3. Positive Elements The Positive Cone in a Hermitian Banach ∗-Algebra The Polar Factorisation of Invertible Elements The Order Structure in a C*-Algebra The Canonical Approximate Unit in a C*-Algebra Quotients and Images of C*-Algebras 57 57 62 65 71 73 Part 2. Representations and Positive Linear Functionals 77 Chapter 4. General Properties of Representations § 22. Continuity Properties of Representations § 23. Cyclic and Non-Degenerate Representations 79 79 85 Chapter 5. States § 24. The GNS Construction 91 91 iii iv CONTENTS § 25. States on Normed ∗-Algebras § 26. Presence of a Unit or Approximate Unit § 27. The Theorem of Varopoulos 97 100 105 Chapter § 28. § 29. § 30. 6. The Enveloping C*-Algebra C*(A) The Theorems of Ra˘ıkov and of Gelfand & Na˘ımark Commutativity